Problem: Compute the exact value of the expression
\[|\pi - |\pi - 7||.\]Write your answer using only integers and $\pi,$ without any absolute value signs.
Explanation: Since $\pi < 7,$
\[|\pi - 7| = 7 - \pi.\]Hence,
\[|\pi - |\pi - 7|| = |\pi - (7 - \pi)| = |2 \pi - 7|.\]We know that $\pi \approx 3.1416 < \frac{7}{2},$ so
\[|2 \pi - 7| = \boxed{7 - 2 \pi}.\]